The Bessel functions are valid for complex argument x,
and an important special case is the situation where x
is purely imaginary: giving a real valued result. In this case the functions
are the two linearly independent solutions to the modified Bessel equation:

The solutions are known as the modified Bessel functions of the first and
second kind (or occasionally as the hyperbolic Bessel functions of the
first and second kind). They are denoted Iv and Kv
respectively: